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  <title>Complex Portfolio Optimization with PortfolioAnalytics</title>
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    <!-- LOGO SLIDE -->
        <slide class="title-slide segue nobackground">
  <hgroup class="auto-fadein">
    <h1>Complex Portfolio Optimization with PortfolioAnalytics</h1>
    <h2>R/Finance 2014</h2>
    <p>Ross Bennett<br/></p>
  </hgroup>
  <article></article>  
</slide>
    

    <!-- SLIDES -->
    <slide class="" id="slide-1" style="background:;">
  <hgroup>
    <h2>Overview</h2>
  </hgroup>
  <article data-timings="">
    <ul>
<li>Discuss Portfolio Optimization</li>
<li>Introduce PortfolioAnalytics</li>
<li>Demonstrate PortfolioAnalytics with Examples</li>
</ul>

<!--
* Discuss Portfolio Optimization
    * Background and challenges of portfolio theory
* Introduce PortfolioAnalytics
    * What PortfolioAnalytics does and the problems it solves
* Demonstrate PortfolioAnalytics with Examples
    * Brief overview of the examples I will be giving
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-2" style="background:;">
  <hgroup>
    <h2>Modern Portfolio Theory</h2>
  </hgroup>
  <article data-timings="">
    <p>&quot;Modern&quot; Portfolio Theory (MPT) was introduced by Harry Markowitz in 1952.</p>

<p>In general, MPT states that an investor&#39;s objective is to maximize portfolio expected return for a given amount of risk.</p>

<p>General Objectives</p>

<ul>
<li>Maximize a measure of gain per unit measure of risk</li>
<li>Minimize a measure of risk</li>
</ul>

<p>How do we define risk? What about more complex objectives and constraints?</p>

<!--
Several approaches follow the Markowitz approach using mean return as a measure of gain and standard deviation of returns as a measure of risk. This is an academic approach. 
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-3" style="background:;">
  <hgroup>
    <h2>Portfolio Optimization Objectives</h2>
  </hgroup>
  <article data-timings="">
    <ul>
<li>Minimize Risk

<ul>
<li>Volatility</li>
<li>Tail Loss (VaR, ES)</li>
<li>Other Downside Risk Measure</li>
</ul></li>
<li>Maximize Risk Adjusted Return

<ul>
<li>Sharpe Ratio, Modified Sharpe Ratio</li>
<li>Several Others</li>
</ul></li>
<li>Risk Budgets

<ul>
<li>Equal Component Contribution to Risk (i.e. Risk Parity)</li>
<li>Limits on Component Contribution</li>
</ul></li>
<li>Maximize a Utility Function

<ul>
<li>Quadratic, CRRA, etc.</li>
</ul></li>
</ul>

<!--
* Expand on pros/cons of closed-form solvers vs. global solvers and what objectives can be solved.
* The challenge here is knowing what solver to use and the capabilities/limits of the chosen solver. 
* Some of these problems can be formulated as a quadratic or linear programming problem. Constructing the constraint matrix and objective function matrix or vector is not trivial. Limited to the quality of LP and QP solvers available for R. 
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-4" style="background:;">
  <hgroup>
    <h2>PortfolioAnalytics Overview</h2>
  </hgroup>
  <article data-timings="">
    <p>PortfolioAnalytics is an R package designed to provide numerical solutions and visualizations for portfolio optimization problems with complex constraints and objectives.</p>

<ul>
<li>Support for multiple constraint and objective types</li>
<li>An objective function can be any valid R function</li>
<li>Modular constraints and objectives</li>
<li>Support for user defined moment functions</li>
<li>Visualizations</li>
<li>Solver agnostic</li>
<li>Support for parallel computing</li>
</ul>

<!---
The key points to make here are:
* Flexibility
    * The multiple types and modularity of constraints and objectives allows us to add, remove, combine, etc. multiple constraint and objective types very easily.
    * Define an objective as any valid R function
    * Define a function to compute the moments (sample, robust, shrinkage, factor model, GARCH model, etc.)
    * Estimation error is a significant concern with optimization. Having the ability to test different models with different parameters is critical.
* PortfolioAnalytics comes "out of the box" with several constraint types.
* Visualization helps to build intuition about the problem and understand the feasible space of portfolios
* Periodic rebalancing and analyzing out of sample performance will help refine objectives and constraints
* Framework for evaluating portfolios with different sets of objectives and portfolios through time
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-5" style="background:;">
  <hgroup>
    <h2>Support Multiple Solvers</h2>
  </hgroup>
  <article data-timings="">
    <p>Linear and Quadratic Programming Solvers</p>

<ul>
<li>R Optimization Infrastructure (ROI)

<ul>
<li>GLPK (Rglpk)</li>
<li>Symphony (Rsymphony)</li>
<li>Quadprog (quadprog)</li>
</ul></li>
</ul>

<p>Global (stochastic or continuous solvers)</p>

<ul>
<li>Random Portfolios</li>
<li>Differential Evolution (DEoptim)</li>
<li>Particle Swarm Optimization (pso)</li>
<li>Generalized Simulated Annealing (GenSA)</li>
</ul>

<!---
Brief explanation of each solver and what optimization problems are supported
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-6" style="background:;">
  <hgroup>
    <h2>Random Portfolios</h2>
  </hgroup>
  <article data-timings="">
    <p>PortfolioAnalytics has three methods to generate random portfolios.</p>

<ol>
<li>The <strong>sample</strong> method to generate random portfolios is based on an idea by Pat Burns.</li>
<li>The <strong>simplex</strong> method to generate random portfolios is based on a paper by W. T. Shaw.</li>
<li>The <strong>grid</strong> method to generate random portfolios is based on the <code>gridSearch</code> function in the NMOF package.</li>
</ol>

<!--
* Random portfolios allow one to generate an arbitray number of portfolios based on given constraints. Will cover the edges as well as evenly cover the interior of the feasible space. Allows for massively parallel execution.

* The sample method to generate random portfolios is based on an idea by Patrick Burns. This is the most flexible method, but also the slowest, and can generate portfolios to satisfy leverage, box, group, and position limit constraints.

* The simplex method to generate random portfolios is based on a paper by W. T. Shaw. The simplex method is useful to generate random portfolios with the full investment constraint, where the sum of the weights is equal to 1, and min box constraints. Values for min_sum and max_sum of the leverage constraint will be ignored, the sum of weights will equal 1. All other constraints such as the box constraint max, group and position limit constraints will be handled by elimination. If the constraints are very restrictive, this may result in very few feasible portfolios remaining. Another key point to note is that the solution may not be along the vertexes depending on the objective. For example, a risk budget objective will likely place the portfolio somewhere on the interior.

* The grid method to generate random portfolios is based on the gridSearch function in NMOF package. The grid search method only satisfies the min and max box constraints. The min_sum and max_sum leverage constraint will likely be violated and the weights in the random portfolios should be normalized. Normalization may cause the box constraints to be violated and will be penalized in constrained_objective.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-7" style="background:;">
  <hgroup>
    <h2>Comparison of Random Portfolio Methods</h2>
  </hgroup>
  <article data-timings="">
    <p><img src="figures/rp_plot.png" alt=""></p>

<!--
* This chart is a prime candidate for an interactive viz
* The feasible space is computed using the EDHEC data for a long only portfolio with a search size of 2000.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-8" style="background:;">
  <hgroup>
    <h2>Comparison of Random Portfolio Methods (Interactive!)</h2>
  </hgroup>
  <article data-timings="">
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"mean": 0.005891779573171,
"sd": 0.01396902649738 
},
{
 "name": "grid",
"mean": 0.005930627259684,
"sd": 0.01386115764594 
},
{
 "name": "grid",
"mean": 0.005965158536585,
"sd": 0.0138081155123 
} 
]
  
      if(!(opts.type==="pieChart" || opts.type==="sparklinePlus" || opts.type==="bulletChart")) {
        var data = d3.nest()
          .key(function(d){
            //return opts.group === undefined ? 'main' : d[opts.group]
            //instead of main would think a better default is opts.x
            return opts.group === undefined ? opts.y : d[opts.group];
          })
          .entries(data);
      }
      
      if (opts.disabled != undefined){
        data.map(function(d, i){
          d.disabled = opts.disabled[i]
        })
      }
      
      nv.addGraph(function() {
        var chart = nv.models[opts.type]()
          .width(opts.width)
          .height(opts.height)
          
        if (opts.type != "bulletChart"){
          chart
            .x(function(d) { return d[opts.x] })
            .y(function(d) { return d[opts.y] })
        }
          
         
        chart
  .color([ "#1f78b4", "#33a02c", "#e31a1c" ])
  .margin({
 "left":    100 
})
          
        chart.xAxis
  .axisLabel("Risk (std. dev.)")
  .tickFormat(d3.format('0.4f'))

        
        
        chart.yAxis
  .axisLabel("Return")
  .tickFormat(d3.format('0.4f'))
      
       d3.select("#" + opts.id)
        .append('svg')
        .datum(data)
        .transition().duration(500)
        .call(chart);

       nv.utils.windowResize(chart.update);
       return chart;
      });
    };
</script>

<!--
The feasible space is computed using the EDHEC data for a long only portfolio with a search size of 2000.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-9" style="background:;">
  <hgroup>
    <h2>Random Portfolios: Simplex Method</h2>
  </hgroup>
  <article data-timings="">
    <p><img src="figures/fev_plot.png" alt=""></p>

<!--
FEV (Face-Edge-Vertex bias values control how concentrated a portfolio is. This can clearly be seen in the plot. As FEV approaches infinity, the portfolio weight will be concentrated on a single asset. PortfolioAnalytics allows you to specify a vector of fev values for comprehensive coverage of the feasible space. 
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-10" style="background:;">
  <hgroup>
    <h2>Workflow: Specify Portfolio</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">args(portfolio.spec)
</code></pre>

<pre><code>## function (assets = NULL, category_labels = NULL, weight_seq = NULL, 
##     message = FALSE) 
## NULL
</code></pre>

<p>Initializes the portfolio object that holds portfolio level data, constraints, and objectives</p>

<!--
The portfolio object is an S3 object that holds portfolio-level data, constraints, and objectives. The portfolio-level data includes asset names and initial weights, labels to categorize assets, and a sequence of weights for random portfolios. The main argument is assets which can be a character vector (most common use), named numeric vector, or scalar value specifying number of assets.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-11" style="background:;">
  <hgroup>
    <h2>Workflow: Add Constraints</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">args(add.constraint)
</code></pre>

<pre><code>## function (portfolio, type, enabled = TRUE, message = FALSE, ..., 
##     indexnum = NULL) 
## NULL
</code></pre>

<p>Supported Constraint Types</p>

<ul>
<li>Sum of Weights</li>
<li>Box</li>
<li>Group</li>
<li>Factor Exposure</li>
<li>Position Limit</li>
<li>and many more</li>
</ul>

<!--
This adds a constraint object to the portfolio object. Constraints are added to the portfolio object with the add.constraint function. Each constraint added is a separate object and stored in the constraints slot in the portfolio object. In this way, the constraints are modular and one can easily add, remove, or modify the constraints in the portfolio object. Main argument is the type, arguments to the constraint constructor are then passed through the dots (...).
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-12" style="background:;">
  <hgroup>
    <h2>Workflow: Add Objectives</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">args(add.objective)
</code></pre>

<pre><code>## function (portfolio, constraints = NULL, type, name, arguments = NULL, 
##     enabled = TRUE, ..., indexnum = NULL) 
## NULL
</code></pre>

<p>Supported Objective types</p>

<ul>
<li>Return</li>
<li>Risk</li>
<li>Risk Budget</li>
<li>Weight Concentration</li>
</ul>

<!--
Objectives are added to the portfolio object with the add.objective function. Each objective added is a separate object and stored in the objectives slot in the portfolio object. In this way, the objectives are modular and one can easily add, remove, or modify the objective objects. The name argument must be a valid R function. Several functions are available in the PerformanceAnalytics package, but custom user defined functions can be used as objective functions.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-13" style="background:;">
  <hgroup>
    <h2>Workflow: Run Optimization</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">args(optimize.portfolio)
</code></pre>

<pre><code>## function (R, portfolio = NULL, constraints = NULL, objectives = NULL, 
##     optimize_method = c(&quot;DEoptim&quot;, &quot;random&quot;, &quot;ROI&quot;, &quot;pso&quot;, &quot;GenSA&quot;), 
##     search_size = 20000, trace = FALSE, ..., rp = NULL, momentFUN = &quot;set.portfolio.moments&quot;, 
##     message = FALSE) 
## NULL
</code></pre>

<pre><code class="r">args(optimize.portfolio.rebalancing)
</code></pre>

<pre><code>## function (R, portfolio = NULL, constraints = NULL, objectives = NULL, 
##     optimize_method = c(&quot;DEoptim&quot;, &quot;random&quot;, &quot;ROI&quot;), search_size = 20000, 
##     trace = FALSE, ..., rp = NULL, rebalance_on = NULL, training_period = NULL, 
##     trailing_periods = NULL) 
## NULL
</code></pre>

<!--
* Notice the similarity between these two functions. You only have to specify a few additional arguments for the backtesting.

* optimize.portfolio: Main arguments for a single period optimization are the returns (R), portfolio, and optimize_method. We take the portfolio object and parse the constraints and objectives according to the optimization method.

* optimize.portfolio.rebalancing: Supports periodic rebalancing (backtesting) to examine out of sample performance. Helps refine constraints and objectives by analyzing out or sample performance. Essentially a wrapper around optimize.portfolio that handles the time interface.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-14" style="background:;">
  <hgroup>
    <h2>Workflow: Analyze Results</h2>
  </hgroup>
  <article data-timings="">
    <table><thead>
<tr>
<th>Visualization</th>
<th>Data Extraction</th>
</tr>
</thead><tbody>
<tr>
<td>plot</td>
<td>extractObjectiveMeasures</td>
</tr>
<tr>
<td>chart.Concentration</td>
<td>extractStats</td>
</tr>
<tr>
<td>chart.EfficientFrontier</td>
<td>extractWeights</td>
</tr>
<tr>
<td>chart.RiskReward</td>
<td>print</td>
</tr>
<tr>
<td>chart.RiskBudget</td>
<td>summary</td>
</tr>
<tr>
<td>chart.Weights</td>
<td></td>
</tr>
</tbody></table>

<!--
Brief explanation of each function.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-15" style="background:;">
  <hgroup>
    <h2>Stock Data Setup</h2>
  </hgroup>
  <article data-timings="">
    <p>Here we will look at portfolio optimization in the context of stocks.</p>

<ul>
<li>Selection of large cap, mid cap, and small cap stocks from CRSP data</li>
<li>Weekly data from 1/7/1997 to 12/28/2010</li>
<li>15 Large Cap</li>
<li>15 Mid Cap</li>
<li>5 Small Cap</li>
</ul>

<pre><code class="r">equity.data &lt;- cbind(largecap_weekly[,1:15], 
                     midcap_weekly[,1:15], 
                     smallcap_weekly[,1:5])
</code></pre>

<!--
slide comments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-16" style="background:;">
  <hgroup>
    <h2>Distribution of Monthly Returns</h2>
  </hgroup>
  <article data-timings="">
    <p><img src="figures/equity_box.png" alt=""></p>

<!--
slide comments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-17" style="background:;">
  <hgroup>
    <h2>Example 1: Market Neutral Portfolio</h2>
  </hgroup>
  <article data-timings="">
    <p>Here we consider a portfolio of stocks. Our objective is to maximize portfolio return with a target of 0.0015 and minimize portfolio StdDev with a target of 0.02 subject to dollar neutral, beta, box, and position limit constraints.</p>

<!--
This involves combining several constraints. This is an MIQPQC problem that can't be solved by quadprog so we will use random portfolios. Good example of a problem with layering multiple constraints and objectives.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-18" style="background:;">
  <hgroup>
    <h2>Specify Portfolio: Constraints</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">portf.dn &lt;- portfolio.spec(stocks)

# Add constraint such that the portfolio weights sum to 0*
portf.dn &lt;- add.constraint(portf.dn, type=&quot;weight_sum&quot;, 
                                  min_sum=-0.01, max_sum=0.01)

# Add box constraint such that no asset can have a weight of greater than
# 20% or less than -20%
portf.dn &lt;- add.constraint(portf.dn, type=&quot;box&quot;, min=-0.2, max=0.2)

# Add constraint such that we have at most 20 positions
portf.dn &lt;- add.constraint(portf.dn, type=&quot;position_limit&quot;, max_pos=20)

# Add constraint such that the portfolio beta is between -0.25 and 0.25
betas &lt;- t(CAPM.beta(equity.data, market, Rf))
portf.dn &lt;- add.constraint(portf.dn, type=&quot;factor_exposure&quot;, B=betas, 
                           lower=-0.25, upper=0.25)
</code></pre>

<!---
explain the constraints
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-19" style="background:;">
  <hgroup>
    <h2>Specify Portfolio: Objectives</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r"># Add objective to maximize portfolio return with a target of 0.0015
portf.dn.StdDev &lt;- add.objective(portf.dn, type=&quot;return&quot;, name=&quot;mean&quot;,
                                 target=0.0015)

# Add objective to minimize portfolio StdDev with a target of 0.02
portf.dn.StdDev &lt;- add.objective(portf.dn.StdDev, type=&quot;risk&quot;, name=&quot;StdDev&quot;,
                                 target=0.02)
</code></pre>

<!---
explain the objectives, specifically the target argument
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-20" style="background:;">
  <hgroup>
    <h2>Run Optimization</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r"># Generate random portfolios
rp &lt;- random_portfolios(portf.dn, 10000, &quot;sample&quot;)

# Run the optimization
opt.dn &lt;- optimize.portfolio(equity.data, portf.dn.StdDev, 
                               optimize_method=&quot;random&quot;, rp=rp,
                               trace=TRUE)
</code></pre>

<!---
generate a set of random portfolios and then pass directly to optimize.portfolio. Could just specify optimize_method = "random" and will automatically generate for you. Generating the random portfolios separately is useful if you plan to evaluate multiple objectives with the same constraints to avoid regenerating random portfolios. Casual benchmarking revealed that generating the random portfolios was often slower than evaluating the objective function on the set of random portfolios. Don't regenerate the portfolios if you don't have to.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-21" style="background:;">
  <hgroup>
    <h2>Plot Results</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">plot(opt.dn, main=&quot;Dollar Neutral Portfolio&quot;, risk.col=&quot;StdDev&quot;, neighbors=10)
</code></pre>

<p><img src="figures/opt_dn.png" alt=""></p>

<!--
Demonstrates the plot method. Visualize the optimal and feasible portfolios in risk-return space and weights of the optimal portfolios. "Neighbor" portfolios allow you to see where the next closest optimal portfolios are (sorted be the value of the objective function from the optimizer).
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-22" style="background:;">
  <hgroup>
    <h2>EDHEC Data Setup</h2>
  </hgroup>
  <article data-timings="">
    <p>Here we will look at portfolio optimization in the context of portfolio of hedge funds.</p>

<ul>
<li>EDHEC-Risk Alternative Indexes monthly returns from 1/31/1997 to 1/31/2014</li>
</ul>

<table><thead>
<tr>
<th>Relative Value</th>
<th>Directional</th>
</tr>
</thead><tbody>
<tr>
<td>Convertible Arbitrage (CA)</td>
<td>CTA Global (CTAG)</td>
</tr>
<tr>
<td>Equity Market Neutral (EMN)</td>
<td>Emerging Markets (EM)</td>
</tr>
<tr>
<td>Fixed Income Arbitrage (FIA)</td>
<td>Global Macro (GM)</td>
</tr>
</tbody></table>

<pre><code class="r">R &lt;- edhec[,c(&quot;Convertible.Arbitrage&quot;, &quot;Equity.Market.Neutral&quot;, 
              &quot;Fixed.Income.Arbitrage&quot;, 
              &quot;CTA.Global&quot;, &quot;Emerging.Markets&quot;, &quot;Global.Macro&quot;)]
# Abreviate column names for convenience and plotting
colnames(R) &lt;- c(&quot;CA&quot;, &quot;EMN&quot;, &quot;FIA&quot;, &quot;CTAG&quot;, &quot;EM&quot;, &quot;GM&quot;)
</code></pre>

<!--
* Chosen to represent investing in hedge funds with different styles
* Not necessarily investable
* Style definitions
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-23" style="background:;">
  <hgroup>
    <h2>Monthly Returns</h2>
  </hgroup>
  <article data-timings="">
    <p><img src="figures/relative_barvar.png" alt="">
<img src="figures/directional_barvar.png" alt=""></p>

<!--
* Easy to see the relative volatility
* Periods of coincident drawdowns
-->

<!--
---

## Distribution of Monthly Returns
![](figures/edhec_box.png)

slide comments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-24" style="background:;">
  <hgroup>
    <h2>Example 2: Minimum Expected Shortfall</h2>
  </hgroup>
  <article data-timings="">
    <p>Consider an allocation to hedge funds using the EDHEC-Risk Alternative Index as a proxy. This will be an extended example starting with an objective to minimize modified expected shortfall, then add risk budget percent contribution limit, and finally add equal risk contribution limit.</p>

<ul>
<li>Minimize Modified Expected Shortfall</li>
<li>Minimize Modified Expected Shortfall with Risk Budget Limit</li>
<li>Minimize Modified Expected Shortfall with Equal Risk Contribution</li>
</ul>

<!--
Add risk budget objective to minimize concentration of percentage component contribution to risk. Concentration is defined as the Herfindahl Hirschman Index (HHI).

$$ \sum_{i=1}^n x_i^2 $$
-->

<!--
* ES is coherent and is a convex function of the weights
* We are actually using modified ES to take into account the higher moments because the return distribution displays skewness and kurtosis. Modified ES uses the Cornish-Fisher estimate of ES.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-25" style="background:;">
  <hgroup>
    <h2>Specify Initial Portfolio</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r"># Specify an initial portfolio
funds &lt;- colnames(R)
portf.init &lt;- portfolio.spec(funds)

# Add constraint such that the weights sum to 1*
portf.init &lt;- add.constraint(portf.init, type=&quot;weight_sum&quot;, 
                             min_sum=0.99, max_sum=1.01)

# Add box constraint such that no asset can have a weight of greater than
# 40% or less than 5%
portf.init &lt;- add.constraint(portf.init, type=&quot;box&quot;, 
                             min=0.05, max=0.4)

# Add return objective with multiplier=0 such that the portfolio mean
# return is calculated, but does not impact optimization
portf.init &lt;- add.objective(portf.init, type=&quot;return&quot;, 
                            name=&quot;mean&quot;, multiplier=0)
</code></pre>

<!--
Explain the initial portfolio setup
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-26" style="background:;">
  <hgroup>
    <h2>Add Objectives</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r"># Add objective to minimize expected shortfall
portf.minES &lt;- add.objective(portf.init, type=&quot;risk&quot;, name=&quot;ES&quot;)

# Add objective to set upper bound on percentage component contribution
portf.minES.RB &lt;- add.objective(portf.minES, type=&quot;risk_budget&quot;, 
                                name=&quot;ES&quot;, max_prisk=0.3)
# Relax box constraints
portf.minES.RB$constraints[[2]]$max &lt;- rep(1,ncol(R))

# Add objective to minimize concentration of modified ES
# component contribution
portf.minES.EqRB &lt;- add.objective(portf.minES, type=&quot;risk_budget&quot;, 
                                  name=&quot;ES&quot;, min_concentration=TRUE)
# Relax box constraints
portf.minES.EqRB &lt;- add.constraint(portf.minES.EqRB, type=&quot;box&quot;, 
                                   min=0.05, max=1, indexnum=2)
</code></pre>

<!--
Key points here are that we are creating 3 new portfolios by reusing the initial portfolio and we are relaxing the box constraints because we are no longer concerned with controlling weight concentration... we have limits on risk contribution.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-27" style="background:;">
  <hgroup>
    <h2>Run Optimization</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r"># Combine the 3 portfolios
portf &lt;- combine.portfolios(list(minES=portf.minES, 
                                 minES.RB=portf.minES.RB, 
                                 minES.EqRB=portf.minES.EqRB))

# Run the optimization
opt.minES &lt;- optimize.portfolio(R, portf, optimize_method=&quot;DEoptim&quot;, 
                                search_size=5000, trace=TRUE, traceDE=0)
</code></pre>

<!---
Explain how portf is a list of portfolios and passed to optimize.portfolio
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-28" style="background:;">
  <hgroup>
    <h2>Plot in Risk-Return Space</h2>
  </hgroup>
  <article data-timings="">
    <p><img src="figures/opt_minES.png" alt=""></p>

<!--
Note here how DEoptim concentrates its search space on the left hand side, towards the objective....  it spends some time in the upper right, but rapidly moves to concentrating closer to the min-risk boundary
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-29" style="background:;">
  <hgroup>
    <h2>Chart Risk Budgets</h2>
  </hgroup>
  <article data-timings="">
    <p><img src="figures/rb_minES.png" alt="">
<img src="figures/eqrb_minES.png" alt=""></p>

<!--
slide comments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-30" style="background:;">
  <hgroup>
    <h2>Set Rebalancing Parameters and Run Backtest</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r"># Set rebalancing frequency
rebal.freq &lt;- &quot;quarters&quot;

# Training Period
training &lt;- 120

# Trailing Period
trailing &lt;- 72

bt.opt.minES &lt;- optimize.portfolio.rebalancing(R, portf,
                                               optimize_method=&quot;DEoptim&quot;, 
                                               rebalance_on=rebal.freq, 
                                               training_period=training, 
                                               trailing_periods=trailing,
                                               search_size=5000,
                                               traceDE=0)
</code></pre>

<!--
slide comments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-31" style="background:;">
  <hgroup>
    <h2>Min ES Risk Contributions and Weights Through Time</h2>
  </hgroup>
  <article data-timings="">
    <p><img src="figures/risk_minES.png" alt="">
<img src="figures/weights_minES.png" alt=""></p>

<!--
slide comments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-32" style="background:;">
  <hgroup>
    <h2>Min ES Risk Budget Limit Risk Contributions and Weights Through Time</h2>
  </hgroup>
  <article data-timings="">
    <p><img src="figures/risk_minESRB.png" alt="">
<img src="figures/weights_minESRB.png" alt=""></p>

<!--
slide comments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-33" style="background:;">
  <hgroup>
    <h2>Min ES Equal Component Contribution Risk Contributions and Weights Through Time</h2>
  </hgroup>
  <article data-timings="">
    <p><img src="figures/risk_minESEqRB.png" alt="">
<img src="figures/weights_minESEqRB.png" alt=""></p>

<!--
slide comments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-34" style="background:;">
  <hgroup>
    <h2>Min ES Equal Component Contribution Risk Contributions and Weights (interactive!)</h2>
  </hgroup>
  <article data-timings="">
    
<div style='float:left;width:48%;' class='centered'>
  <h4>Percent Contribution to Risk</h4>

<div id = 'chart43e629e31ef' class = 'rChart nvd3'></div>

<script type='text/javascript'>
 $(document).ready(function(){
      drawchart43e629e31ef()
    });
    function drawchart43e629e31ef(){  
      var opts = {
 "dom": "chart43e629e31ef",
"width":    450,
"height":    400,
"x": "date",
"y": "risk",
"group": "fund",
"type": "multiBarChart",
"id": "chart43e629e31ef" 
},
        data = [
 {
 "date":          13513,
"fund": "CA",
"risk": 0.157582705534 
},
{
 "date":          13603,
"fund": "CA",
"risk": 0.2054470067933 
},
{
 "date":          13694,
"fund": "CA",
"risk": 0.172226726396 
},
{
 "date":          13786,
"fund": "CA",
"risk": 0.1403443983496 
},
{
 "date":          13878,
"fund": "CA",
"risk": 0.1654702119764 
},
{
 "date":          13969,
"fund": "CA",
"risk": 0.1644295540582 
},
{
 "date":          14060,
"fund": "CA",
"risk": 0.1722157951409 
},
{
 "date":          14152,
"fund": "CA",
"risk": 0.1596198199404 
},
{
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"fund": "CA",
"risk": 0.1585109419483 
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{
 "date":          14334,
"fund": "CA",
"risk": 0.1573087197099 
},
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 "date":          14425,
"fund": "CA",
"risk": 0.1776582080179 
},
{
 "date":          14517,
"fund": "CA",
"risk": 0.1662894963767 
},
{
 "date":          14609,
"fund": "CA",
"risk": 0.1637173004128 
},
{
 "date":          14699,
"fund": "CA",
"risk": 0.1762879302251 
},
{
 "date":          14790,
"fund": "CA",
"risk": 0.1240208291949 
},
{
 "date":          14882,
"fund": "CA",
"risk": 0.1664940300598 
},
{
 "date":          14974,
"fund": "CA",
"risk": 0.1700949293145 
},
{
 "date":          15064,
"fund": "CA",
"risk": 0.1492801731167 
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<div style='float:right;width:48%;'>
  <h4>Weights</h4>

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]
  
      if(!(opts.type==="pieChart" || opts.type==="sparklinePlus" || opts.type==="bulletChart")) {
        var data = d3.nest()
          .key(function(d){
            //return opts.group === undefined ? 'main' : d[opts.group]
            //instead of main would think a better default is opts.x
            return opts.group === undefined ? opts.y : d[opts.group];
          })
          .entries(data);
      }
      
      if (opts.disabled != undefined){
        data.map(function(d, i){
          d.disabled = opts.disabled[i]
        })
      }
      
      nv.addGraph(function() {
        var chart = nv.models[opts.type]()
          .width(opts.width)
          .height(opts.height)
          
        if (opts.type != "bulletChart"){
          chart
            .x(function(d) { return d[opts.x] })
            .y(function(d) { return d[opts.y] })
        }
          
         
        chart
  .color([ "#a6cee3", "#1f78b4", "#b2df8a", "#33a02c", "#fb9a99", "#e31a1c" ])
  .stacked(true)
          
        chart.xAxis
  .tickFormat(function(d){
    return d3.time.format('%b %Y')(new Date(d * 24 * 60 * 60 * 1000))
    })

        
        
        chart.yAxis
  .tickFormat(function(d){
    return d3.format('0.2%')(d)
    })
      
       d3.select("#" + opts.id)
        .append('svg')
        .datum(data)
        .transition().duration(500)
        .call(chart);

       nv.utils.windowResize(chart.update);
       return chart;
      });
    };
</script>

<!--
slide comments
-->

</div>
  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-35" style="background:;">
  <hgroup>
    <h2>Compute Returns</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">ret.bt.opt &lt;- do.call(cbind, lapply(bt.opt.minES, 
                                    function(x) summary(x)$portfolio_returns))
colnames(ret.bt.opt) &lt;- c(&quot;min ES&quot;, &quot;min ES RB&quot;, &quot;min ES Eq RB&quot;)
</code></pre>

<!--
Recall that bt.opt.minES is a list where each element is an optimize.portfolio.rebalancing object. Basically, we want to loop through that list and extract the portfolio returns that is computed by the summary method.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-36" style="background:;">
  <hgroup>
    <h2>Chart Performance</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">charts.PerformanceSummary(ret.bt.opt)
</code></pre>

<p><img src="figures/ret_minES.png" alt=""></p>

<!--
slide comments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-37" style="background:;">
  <hgroup>
    <h2>Example 3: Maximize CRRA</h2>
  </hgroup>
  <article data-timings="">
    <p>Consider an allocation to hedge funds using the EDHEC-Risk Alternative Index as a proxy. Our objective to maximize the fourth order expansion of the Constant Relative Risk Aversion (CRRA) expected utility function as in the Boudt paper and Martellini paper. We use the same data as Example 3.</p>

<p>\[
EU_{\lambda}(w) = - \frac{\lambda}{2} m_{(2)}(w) + 
\frac{\lambda (\lambda + 1)}{6} m_{(3)}(w) -
\frac{\lambda (\lambda + 1) (\lambda + 2)}{24} m_{(4)}(w)
\]</p>

<!---
* This example will demonstrate a custom moment function and a custom objective function.
* This is an interesting objective function
* An increase in the 2nd and 4th moment will decrease the objective function
* An increase in the lambda value places a bigger penalty on the higher moments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-38" style="background:;">
  <hgroup>
    <h2>Define a function to compute CRRA</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">CRRA &lt;- function(R, weights, lambda, sigma, m3, m4){
  weights &lt;- matrix(weights, ncol=1)
  M2.w &lt;- t(weights) %*% sigma %*% weights
  M3.w &lt;- t(weights) %*% m3 %*% (weights %x% weights)
  M4.w &lt;- t(weights) %*% m4 %*% (weights %x% weights %x% weights)
  term1 &lt;- (1 / 2) * lambda * M2.w
  term2 &lt;- (1 / 6) * lambda * (lambda + 1) * M3.w
  term3 &lt;- (1 / 24) * lambda * (lambda + 1) * (lambda + 2) * M4.w
  out &lt;- -term1 + term2 - term3
  out
}
</code></pre>

<!---
The function arguments should have 'R' as the name of the returns and 'weights' as the name of the weights. 'R' and 'weights' are automatically matched, any other function arguments can be passed in through arguments in add.objective.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-39" style="background:;">
  <hgroup>
    <h2>Define a custom moment function</h2>
  </hgroup>
  <article data-timings="">
    <p>The default function for <code>momentFUN</code> is <code>set.portfolio.moments</code>. We need to write our own function to estimate the moments for our objective function.</p>

<pre><code class="r">crra.moments &lt;- function(R, ...) {
    out &lt;- list()
    out$mu &lt;- colMeans(R)
    out$sigma &lt;- cov(R)
    out$m3 &lt;- PerformanceAnalytics:::M3.MM(R)
    out$m4 &lt;- PerformanceAnalytics:::M4.MM(R)
    out
}
</code></pre>

<!--
The set.portfolio.moments looks for the most common objective function names e.g. mean, StdDev, VaR, and ES. The custom moment function should return a named list of the moments needed for the objective function.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-40" style="background:;">
  <hgroup>
    <h2>Specify Portfolio</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r"># Specify portfolio
portf.crra &lt;- portfolio.spec(funds)

# Add constraint such that the weights sum to 1
portf.crra &lt;- add.constraint(portf.crra, type=&quot;weight_sum&quot;, 
                             min_sum=0.99, max_sum=1.01)

# Add box constraint such that no asset can have a weight of greater than
# 40% or less than 5% 
portf.crra &lt;- add.constraint(portf.crra, type=&quot;box&quot;, 
                             min=0.05, max=0.4)

# Add objective to maximize CRRA
portf.crra &lt;- add.objective(portf.crra, type=&quot;return&quot;, 
                            name=&quot;CRRA&quot;, arguments=list(lambda=10))
</code></pre>

<!---
Focus on how CRRA is added as an objective and the lambda argument is passed
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-41" style="background:;">
  <hgroup>
    <h2>&quot;Dummy&quot; Objectives</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r"># Dummy objectives for plotting and/or further analysis
portf.crra &lt;- add.objective(portf.crra, type=&quot;return&quot;, name=&quot;mean&quot;, multiplier=0)
portf.crra &lt;- add.objective(portf.crra, type=&quot;risk&quot;, name=&quot;ES&quot;, multiplier=0)
portf.crra &lt;- add.objective(portf.crra, type=&quot;risk&quot;, name=&quot;StdDev&quot;, multiplier=0)
</code></pre>

<!---
* Explain what multiplier = 0 is doing
* The value is computed, but it does not impact the optimization
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-42" style="background:;">
  <hgroup>
    <h2>Run Optimization</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">opt.crra &lt;- optimize.portfolio(R, portf.crra, optimize_method=&quot;DEoptim&quot;, 
                                 search_size=5000, trace=TRUE, traceDE=0,
                                 momentFUN=&quot;crra.moments&quot;)
</code></pre>

<pre><code class="r">head(extractStats(opt.crra),4)
</code></pre>

<pre><code>##              CRRA.CRRA     mean      ES  StdDev       out   w.CA  w.EMN
## .DE.portf.1 -0.0009786 0.005906 0.03407 0.01367 0.0009786 0.1667 0.1667
## .DE.portf.2 -0.0007585 0.005716 0.02532 0.01215 0.0007585 0.1560 0.3420
## .DE.portf.3 -0.0019451 0.006380 0.05068 0.01901 0.0019451 0.1180 0.1200
## .DE.portf.4 -0.0007816 0.005536 0.02591 0.01232 0.0007816 0.0900 0.1840
##              w.FIA w.CTAG   w.EM   w.GM
## .DE.portf.1 0.1667 0.1667 0.1667 0.1667
## .DE.portf.2 0.0560 0.2180 0.1280 0.0920
## .DE.portf.3 0.0880 0.1300 0.3900 0.1620
## .DE.portf.4 0.2800 0.2720 0.1160 0.0640
</code></pre>

<!--
remember to specify a momentFUN to match the arguments in CRRA
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-43" style="background:;">
  <hgroup>
    <h2>Chart Results</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">chart.RiskReward(opt.crra, risk.col = &quot;ES&quot;)
chart.RiskReward(opt.crra, risk.col = &quot;StdDev&quot;)
</code></pre>

<p><img src="figures/crra_RR_ES.png" alt="">
<img src="figures/crra_RR_StdDev.png" alt=""></p>

<!--
It is very interesting to look at the feasible space in terms of different measures for risk. It appears that the optimal CRRA portfolio is close to the minimum StdDev portfolio, but nowhere near the minimum ES portfolio.
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-44" style="background:;">
  <hgroup>
    <h2>Run Backtest and Compute Returns</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">bt.opt.crra &lt;- optimize.portfolio.rebalancing(R, portf.crra, 
                                              optimize_method=&quot;DEoptim&quot;,
                                              search_size=5000, trace=TRUE,
                                              traceDE=0,
                                              momentFUN=&quot;crra.moments&quot;,
                                              rebalance_on=rebal.freq, 
                                              training_period=training, 
                                              trailing_periods=trailing)

ret.crra &lt;- summary(bt.opt.crra)$portfolio_returns
colnames(ret.crra) &lt;- &quot;CRRA&quot;
</code></pre>

<!---
Run optimization and extract the portfolio rebalancing returns from the summary method. 
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-45" style="background:;">
  <hgroup>
    <h2>Chart Weights Through Time</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">chart.Weights(bt.opt.crra, main=&quot;CRRA Weights&quot;, col=bluemono)
</code></pre>

<p><img src="figures/weights_crra.png" alt=""></p>

<!--
slide comments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-46" style="background:;">
  <hgroup>
    <h2>Chart Performance</h2>
  </hgroup>
  <article data-timings="">
    <pre><code class="r">charts.PerformanceSummary(cbind(ret.bt.opt, ret.crra), 
                          main=&quot;Optimization Performance&quot;)
</code></pre>

<p><img src="figures/ret_crra.png" alt=""></p>

<!--
slide comments
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-47" style="background:;">
  <hgroup>
    <h2>Conclusion</h2>
  </hgroup>
  <article data-timings="">
    <ul>
<li>Introduced the goals and summary of PortfolioAnalytics</li>
<li>Demonstrated the flexibility through examples</li>
<li>Exciting plans for GSOC 2014

<ul>
<li>Support for regime switching</li>
<li>Support for supervised learning</li>
<li>many more</li>
</ul></li>
</ul>

<h4>Acknowledgements</h4>

<p>Many thanks to...</p>

<ul>
<li>Google: funding for Google Summer of Code (GSoC)</li>
<li>UW CF&amp;RM Program: continued work on PortfolioAnalytics</li>
<li>GSoC Mentors: Brian Peterson, Peter Carl, Doug Martin, and Guy Yollin</li>
<li>R/Finance Committee</li>
</ul>

<!---
- One of the best things about GSoC is the opportunity to work and interact with the mentors.
- Thank the GSoC mentors for offering help and guidance during the GSoC project and after as I continued to work on the PortfolioAnalytics package.
- R/Finance Committee for the conference and the opportunity to talk about PortfolioAnalytics.
- Google for funding the Google Summer of Code for PortfolioAnalytics and many other proposals for R
-->

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-48" style="background:;">
  <hgroup>
    <h2>PortfolioAnalytics Links</h2>
  </hgroup>
  <article data-timings="">
    <p>PortfolioAnalytics is on R-Forge in the ReturnAnalytics project</p>

<ul>
<li><a href="https://r-forge.r-project.org/projects/returnanalytics/">PortfolioAnalytics</a></li>
</ul>

<p>Source code for the slides</p>

<ul>
<li><a href="https://github.com/rossb34/PortfolioAnalyticsPresentation">https://github.com/rossb34/PortfolioAnalyticsPresentation</a></li>
</ul>

<p>and view it here</p>

<ul>
<li><a href="http://rossb34.github.io/PortfolioAnalyticsPresentation/">http://rossb34.github.io/PortfolioAnalyticsPresentation/</a></li>
</ul>

  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-49" style="background:;">
  <hgroup>
    <h2>Any Questions?</h2>
  </hgroup>
  <article data-timings="">
    
  </article>
  <!-- Presenter Notes -->
</slide>

<slide class="" id="slide-50" style="background:;">
  <hgroup>
    <h2>References and Useful Links</h2>
  </hgroup>
  <article data-timings="">
    <ul>
<li><a href="http://cran.r-project.org/web/packages/ROI/index.html">ROI</a></li>
<li><a href="http://cran.r-project.org/web/packages/DEoptim/index.html">DEoptim</a></li>
<li><a href="http://cran.r-project.org/web/packages/pso/index.html">pso</a></li>
<li><a href="http://cran.r-project.org/web/packages/GenSA/index.html">GenSA</a></li>
<li><a href="http://cran.r-project.org/web/packages/PerformanceAnalytics/index.html">PerformanceAnalytics</a></li>
<li><a href="http://www.burns-stat.com/pages/Finance/randport_practice_theory_annotated.pdf">Patrick Burns Random Portfolios</a></li>
<li><a href="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1856476">W.T. Shaw Random Portfolios</a></li>
<li>Martellini paper</li>
<li>Boudt paper</li>
<li><a href="http://spark.rstudio.com/rossbennett3/PortfolioOptimization/">Shiny App</a></li>
</ul>

  </article>
  <!-- Presenter Notes -->
</slide>

    <slide class="backdrop"></slide>
  </slides>
  <div class="pagination pagination-small" id='io2012-ptoc' style="display:none;">
    <ul>
      <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=1 title='Overview'>
         1
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=2 title='Modern Portfolio Theory'>
         2
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=3 title='Portfolio Optimization Objectives'>
         3
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=4 title='PortfolioAnalytics Overview'>
         4
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=5 title='Support Multiple Solvers'>
         5
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=6 title='Random Portfolios'>
         6
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=7 title='Comparison of Random Portfolio Methods'>
         7
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=8 title='Comparison of Random Portfolio Methods (Interactive!)'>
         8
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=9 title='Random Portfolios: Simplex Method'>
         9
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=10 title='Workflow: Specify Portfolio'>
         10
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=11 title='Workflow: Add Constraints'>
         11
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=12 title='Workflow: Add Objectives'>
         12
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=13 title='Workflow: Run Optimization'>
         13
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=14 title='Workflow: Analyze Results'>
         14
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=15 title='Stock Data Setup'>
         15
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=16 title='Distribution of Monthly Returns'>
         16
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=17 title='Example 1: Market Neutral Portfolio'>
         17
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=18 title='Specify Portfolio: Constraints'>
         18
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=19 title='Specify Portfolio: Objectives'>
         19
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=20 title='Run Optimization'>
         20
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=21 title='Plot Results'>
         21
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=22 title='EDHEC Data Setup'>
         22
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=23 title='Monthly Returns'>
         23
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=24 title='Example 2: Minimum Expected Shortfall'>
         24
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=25 title='Specify Initial Portfolio'>
         25
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=26 title='Add Objectives'>
         26
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=27 title='Run Optimization'>
         27
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=28 title='Plot in Risk-Return Space'>
         28
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=29 title='Chart Risk Budgets'>
         29
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=30 title='Set Rebalancing Parameters and Run Backtest'>
         30
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=31 title='Min ES Risk Contributions and Weights Through Time'>
         31
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=32 title='Min ES Risk Budget Limit Risk Contributions and Weights Through Time'>
         32
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=33 title='Min ES Equal Component Contribution Risk Contributions and Weights Through Time'>
         33
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=34 title='Min ES Equal Component Contribution Risk Contributions and Weights (interactive!)'>
         34
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=35 title='Compute Returns'>
         35
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=36 title='Chart Performance'>
         36
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=37 title='Example 3: Maximize CRRA'>
         37
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=38 title='Define a function to compute CRRA'>
         38
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=39 title='Define a custom moment function'>
         39
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=40 title='Specify Portfolio'>
         40
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=41 title='&quot;Dummy&quot; Objectives'>
         41
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=42 title='Run Optimization'>
         42
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=43 title='Chart Results'>
         43
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=44 title='Run Backtest and Compute Returns'>
         44
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=45 title='Chart Weights Through Time'>
         45
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=46 title='Chart Performance'>
         46
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=47 title='Conclusion'>
         47
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=48 title='PortfolioAnalytics Links'>
         48
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=49 title='Any Questions?'>
         49
      </a>
    </li>
    <li>
      <a href="#" target="_self" rel='tooltip' 
        data-slide=50 title='References and Useful Links'>
         50
      </a>
    </li>
  </ul>
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